Department of Applied Physics and Applied Mathematics
Columbia University
APPH E4210. Geophysical Fluid Dynamics
Spring 2008
Problem Set 3
(Due Feb 21, 2008)
1. This problem is based on the film “Waves across the Pacific” which you should have seen
beforehand. The film is about a field experiment carried out during the 1960’s to track surface
gravity waves generated by storms in the Indian Ocean, off New Zealand, as they propagate
across the Pacific Ocean all the way to Alaska. In the experiment, an array of ocean bottom
pressure gauges were placed along a great circle route from Antarctica to Alaska (see figure
below taken from Snodgrass et al, Phil. Trans. A (1966)). The pressure gauges measured
fluctuations in sea surface height due to ocean swells to a phenomenal precision of 0.1 mm!
(The experiment involved much arduous field work on remote Pacific islands, including
Samoa and Hawaii.)
One of the principal goals of the experiment was to use the dispersive property of surface
waves, i.e., the fact that group velocity is a function of wavenumber, to locate their point
of origin. (If the distance from the source region is large, then the source can be treated as
a single point.) At a fixed point in space, dispersion manifests itself as a change over time
in the observed frequency of waves. A beautiful example of this phenomenon is evident
in the figure below. The top panel shows the observed energy spectrum of a wave train at
Honolulu at roughly 12 hour intervals. (Ignore the lower two panels.) At any instant, there
is a peak in the spectrum corresponding to the dominant frequency of the wave group. This
peak shifts to higher frequencies over time. (The frequency is in millicycles per second or
mHz.) Use the information in the plot to make a rough estimate of the distance of the wave
source from Honolulu. Give your answer in degrees. To simplify the calculation, recall
that the wavelength of these waves is much smaller than the depth of the ocean, i.e., we are
considering “deep water” waves.
2. Geostrophic adjustment. Consider the shallow water equations on an f -plane. Suppose that
at t = 0, the velocity field is zero and the surface elevation is given by
η = ηo , −a ≤ y ≤ a,
and zero elsewhere.
(a) Write down the appropriate Klein-Gordon equation governing the time evolution of η.
(b) Write the solution as the sum of a time-dependent homogeneous solution (ηh ) and a
steady particular solution (ηs ). Find the steady, geostrophic solution ηs . Hint: You
should end up with a 2d order inhomogeneous ODE. To derive the appropriate “boundary conditions”, integrate this equation over a small interval centered about ±a, and
then let the interval go to zero. This establishes the continuity (or lack thereof) of η
and dη/dy at ±a. The change in η or its derivative across ±a is known as a “jump
condition” and provides us with the necessary constraints.
(c) Use the momentum equations to find the geostrophic velocity field.
(d) Compute the ratio R of the total energy in the final geostrophic state to that in the
initial state. Express, and make a plot of, this ratio as a function of a/λd , where λd is
the deformation radius.
(e) Transient solution. Now that you have found the steady (particular) solution, lets calculate the time-dependent (homogeneous) solution. While this transient solution can be
found analytically by means of Fourier or Laplace transforms, the inverse transforms
are difficult to work out (and not so readily found in tables either). Here, I walk you
through the steps necessary to obtain the solution numerically. The basic idea is to use
the discrete Fourier transform (implemented as the FFT in numerical software such as
MATLAB) to do the inverse transform.
i. Write down the PDE for the homogeneous part, ηh , and the initial conditions it is
subject to.
ii. Take the Fourier transform of the equation (in the spatial direction) and solve the
resulting ODE (for the Fourier transform). Equivalently, you could simply write
down the solution as a sum of left- and right-going plane wave solutions using the
known (Poincare) dispersion relation for this equation.
iii. Use the DFT (FFT) to invert the transform solution back into physical space.
iv. Make plots of the full time dependent solution η = ηh +ηs at several times showing
the approach to a steady state. If you nondimensionalize time and space appropriately, your life will be greatly simplified.